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Calculus Solver

Implicit Differentiation Calculator

Enter an equation in x and y, such as x² + y² = 25 or x³ + y³ = 6xy, to find dy/dx. Add a point on the curve to get the slope and the equation of the tangent line there.

Your expression will be shown here as it will be read.

Finds dy/dx. Add a point on the curve to get the slope and tangent line there.

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What is implicit differentiation?

Some curves are defined by an equation that is not solved for y, such as the circle x² + y² = 25. Solving for y can be awkward or impossible, but you can still find the slope dy/dx by treating y as an unknown function of x.

Differentiate both sides of the equation with respect to x. Every time you differentiate a term containing y, the chain rule attaches a factor dy/dx. Then solve the resulting equation for dy/dx. The answer usually contains both x and y, which is expected.

F(x,y)=0⟹dydx=−FxFyF(x, y) = 0 \quad\Longrightarrow\quad \frac{dy}{dx} = -\frac{F_{x}}{F_{y}}

The rules you need

Implicit differentiation uses the ordinary rules, plus one idea: y is a function of x.

  • Powers of y

    ddx[yn]=n yn−1dydx\frac{d}{dx}\left[y^{n}\right] = n\,y^{n-1}\frac{dy}{dx}

    The chain rule, with y as the inner function.

  • Products of x and y

    ddx[xy]=y+xdydx\frac{d}{dx}\left[xy\right] = y + x\frac{dy}{dx}

    Use the product rule; the y factor contributes dy/dx.

  • Solve for dy/dx

    dydx(…)=…\frac{dy}{dx}\big(\ldots\big) = \ldots

    Move every term containing dy/dx to one side, factor it out, and divide.

  • Formula method

    dydx=−∂F/∂x∂F/∂y\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}

    Write the equation as F(x, y) = 0. This gives the same answer as differentiating both sides.

  • Horizontal and vertical tangents

    dydx=0 (horizontal),Fy=0 (vertical)\frac{dy}{dx} = 0 \ \text{(horizontal)},\qquad F_{y} = 0 \ \text{(vertical)}

    Horizontal tangents occur where the numerator is zero; vertical tangents where the denominator is zero.

How this calculator works

The calculator moves every term to one side to get F(x, y) = 0, then computes the partial derivatives F_x (holding y constant) and F_y (holding x constant), showing each rule applied. It combines them as dy/dx = −F_x / F_y, simplifies, and cancels common numerical factors.

If you give a point, the calculator first checks that it lies on the curve, then substitutes it exactly to get the slope and the tangent line. If F_y is zero there, it reports a vertical tangent instead of dividing by zero.

The answer is checked numerically: the calculator finds points on the curve, follows the curve a tiny step either side of each one, and confirms that the slope it measures matches the formula.

For the full details, including how answers are verified, see how the solver works.

Worked examples

Work through these by hand first, then enter them above to compare each step.

Example 1

The circle

x2+y2=25x^{2} + y^{2} = 25

  1. Differentiate both sides with respect to x; y² gives 2y·dy/dx by the chain rule.

    2x+2ydydx=02x + 2y\frac{dy}{dx} = 0

  2. Solve for dy/dx.

    dydx=−2x2y\frac{dy}{dx} = -\frac{2x}{2y}

  3. At the point (3, 4) the slope is −3/4, so the tangent line is y = −(3/4)x + 25/4.

Answer

dydx=−xy\frac{dy}{dx} = -\frac{x}{y}

Example 2

The folium of Descartes

x3+y3=6xyx^{3} + y^{3} = 6xy

  1. Differentiate both sides; the right side needs the product rule.

    3x2+3y2dydx=6y+6xdydx3x^{2} + 3y^{2}\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}

  2. Collect the dy/dx terms and factor.

    dydx(3y2−6x)=6y−3x2\frac{dy}{dx}\left(3y^{2} - 6x\right) = 6y - 3x^{2}

  3. Divide, and cancel the common factor 3.

Answer

dydx=2y−x2y2−2x\frac{dy}{dx} = \frac{2y - x^{2}}{y^{2} - 2x}

Example 3

Product rule terms

x2y+y3=4xx^{2}y + y^{3} = 4x

  1. Product rule on x²y, chain rule on y³:

    2xy+x2dydx+3y2dydx=42xy + x^{2}\frac{dy}{dx} + 3y^{2}\frac{dy}{dx} = 4

  2. Factor out dy/dx:

    dydx(x2+3y2)=4−2xy\frac{dy}{dx}\left(x^{2} + 3y^{2}\right) = 4 - 2xy

Answer

dydx=4−2xyx2+3y2\frac{dy}{dx} = \frac{4 - 2xy}{x^{2} + 3y^{2}}

Example 4

A trigonometric equation

sin⁡(xy)=x\sin(xy) = x

  1. Chain rule on the left; the inner function xy needs the product rule.

    cos⁡(xy)(y+xdydx)=1\cos(xy)\left(y + x\frac{dy}{dx}\right) = 1

  2. Expand and solve for dy/dx.

    xcos⁡(xy)dydx=1−ycos⁡(xy)x\cos(xy)\frac{dy}{dx} = 1 - y\cos(xy)

Answer

dydx=1−ycos⁡(xy)xcos⁡(xy)\frac{dy}{dx} = \frac{1 - y\cos(xy)}{x\cos(xy)}

Example 5

An exponential equation

ey+xy=ee^{y} + xy = e

  1. Differentiate: eʸ gives eʸ·dy/dx, and xy gives y + x·dy/dx.

    eydydx+y+xdydx=0e^{y}\frac{dy}{dx} + y + x\frac{dy}{dx} = 0

  2. Solve for dy/dx. At the point (0, 1), this gives −1/e.

Answer

dydx=−yx+ey\frac{dy}{dx} = -\frac{y}{x + e^{y}}

Common mistakes

  • Forgetting dy/dx on y terms

    d/dx[y²] is 2y·dy/dx, not 2y. Every term containing y needs the chain rule factor.

  • Missing the product rule on xy

    d/dx[xy] = y + x·dy/dx. Writing just dy/dx, or just y, loses a term.

  • Differentiating the constant side

    In x² + y² = 25, the right side becomes 0. Constants on either side always differentiate to zero.

  • Plugging in a point that's not on the curve

    The slope formula only means something at points that satisfy the equation. Check the point first; the calculator does this for you.

Limitations

  • Finds dy/dx with x as the independent variable and y as the dependent one. For dx/dy, swap the roles of x and y in your equation.
  • Second derivatives d²y/dx² of implicit equations are not computed.
  • Other letters (such as a or r) are treated as constants.
  • The numerical check needs real points on the curve between x = −20 and 20; if none are found, the result is shown without that check.

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Frequently asked questions

When do you use implicit differentiation?

When y is not given explicitly as a function of x, as in x² + y² = 25 or x³ + y³ = 6xy. It is also used for related rates problems and for proving derivative formulas such as that of arctan.

Why does the answer contain y?

Because y depends on x through the equation, the slope at a point depends on both coordinates. On the circle x² + y² = 25, the points (3, 4) and (3, −4) have different slopes, which −x/y captures.

How do I find the slope at a specific point?

Enter the point in the “At the point” box, for example 3, 4. The calculator checks the point is on the curve, then gives the slope and the tangent line there.

Is the formula −F_x/F_y the same as differentiating both sides?

Yes. Both methods give the same dy/dx. The worked examples on this page show the both-sides method; the calculator's steps use the formula, which shows each partial derivative clearly.

Next step

Pick one of the worked examples, solve it on paper, then enter it in the calculator and compare your steps with each rule shown. Spotting where your work differs is the fastest way to improve.

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